The matrix that is a graph
A matrix with no numbers in it
A graph arrives as vertices and edges. Two different matrices can be built from it, they answer different questions, and one of them has a null vector that is exact — the only object in these essays whose kernel is known before anything runs.
Two Laplacians of one graph
The combinatorial Laplacian D − A and the normalised one, which conjugates it by the inverse square roots of the degrees, are built from the same object, are not similar to each other, and answer different questions. On a graph whose degrees are equal they coincide. On one whose degrees span an order of magnitude their second eigenvalues are sixteen times apart.
The vertex nobody solves for
A Laplacian is singular, so every solve with one has to remove its kernel first. There are three ways, they agree to fourteen digits, and the one everybody uses carries a free parameter that no account of the method mentions and that moves the condition number by nine hundred.
The vector that has to be rounded
A spectral partition is an eigenvector, and an eigenvector is a real vector. The answer wanted is a subset. Something has to turn one into the other, and the something is a heuristic applied after the linear algebra has finished.
A bound with a square root in it
Cheeger's inequality brackets a graph's best cut between λ₂/2 and √(2λ₂). The lower bound is attained exactly. The upper one is loose by a factor of fourteen — on the one graph in the census with a real bottleneck, which is the shape it is always quoted about.
A partition decided in the last digit
On a graph with a symmetry there is no Fiedler vector — there is a plane, and every vector in it is an exact eigenvector. Twenty-four runs with the edge weights nudged by 10⁻¹² return ten different partitions of a cycle and, on a hypercube, two different qualities of answer.
A ranking that is an eigenvector
PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.
The rate is the second eigenvalue
A walk forgets where it started at a rate the graph's second eigenvalue names exactly. Across three orders of magnitude in the step count the prediction is five per cent high — and the published rate for PageRank is right for a reason nobody states, which is that a link graph is in pieces.
A chain with no stationary vector
A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.
An eigenvector that must not change sign
Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the only two-sided bound in these essays that is proved rather than estimated.
A ranking whose order is not determined
Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.
A distance computed by a solve
Effective resistance is the one quantity in this field with no combinatorial route to it — it is defined by a linear system. On a small unweighted graph the answer is a ratio of two integers, so for once the error is known rather than estimated, and every resistance in a graph has to add up to a number fixed in advance.
A graph with a tenth of the edges
Keeping 344 of 1,225 edges, sampled by effective resistance and reweighted, preserves every eigenvalue of the Laplacian to within a factor of 1.7. It preserves no degree — half of them are wrong by more than a third — and it takes the diameter from one to three.
Eliminating a vertex is a graph operation
Gaussian elimination on a Laplacian deletes a vertex and joins its neighbours into a clique with conductances wᵢwⱼ over Σw. The matrix that remains is still a graph — symmetric, zero row sums, nonpositive off the diagonal — and the ordering decides whether the fill is thirty-one edges or four hundred and sixty-five.
A preconditioner that is a tree
Every eigenvalue of a tree-preconditioned Laplacian is at least one and at most the total stretch — a combinatorial integer with no arithmetic in it. Measured, the bound is two to four times loose, and on a grid the preconditioner makes the conditioning worse by a factor of 1.85 at every size.
A count that comes out of a determinant
The number of spanning trees of a graph is the determinant of its grounded Laplacian, so it is a whole number known in advance. The elimination that computes it is backward stable at every size — and from sixteen vertices the answer is wrong, because the count has seventeen digits and a binary64 has sixteen.
The spectrum is not the graph
Two graphs on six vertices with the same Laplacian characteristic polynomial — as integer polynomials, not to fourteen digits. One contains a triangle; the other is bipartite. Every method in this field that reads only the spectrum is answering about the class.
A Laplacian that is not symmetric
Point the edges and the matrix stops being symmetric. Its row sums are still exactly zero, so the null vector survives; everything built on the quadratic form does not, and the eigenvalues need a plane rather than a line. Asymmetry permits that and does not force it, which the smallest case here demonstrates by being asymmetric and real.
A conductance the arcs do not measure
Symmetrising a directed Laplacian with respect to its walk recovers everything the arrows took — a real spectrum, a sweep cut, a Cheeger inequality. What it does not recover is the quantity: the inequality bounds the probability that a step of the walk crosses the cut, which on one graph here is three times the weight of the arcs that do.
A finer invariant that hears less
The Laplacian spectrum fixes a graph's number of spanning trees and not the group those trees form, so the Smith normal form of the grounded Laplacian is a strictly finer integer invariant — and on the six-vertex pair the spectrum cannot separate, it does: ℤ₁₂ against ℤ₂ ⊕ ℤ₆. Over all 1,022 Laplacian-cospectral pairs of connected graphs on eight vertices it separates 435. The adjacency polynomial, a second spectrum rather than a finer one, separates all of them.
Each spectrum hears the other's pairs
Pairs of graphs with the same Laplacian spectrum share a spanning-tree count, so the critical group could only tell them apart by how that count factors — and did on two pairs in five. Turn the census round, to the 733 pairs of eight-vertex graphs with the same adjacency spectrum, and the tree count is free to differ. It does on 717 of them, by a median of six per cent. The critical group separates exactly those 717 and not one more: on all sixteen pairs whose counts agree, including a non-cyclic one, the two graphs have the same group. And the Laplacian spectrum separates all 733, as the adjacency spectrum separated every Laplacian pair.
The group hears a switch only at two
On a regular graph the Laplacian is the degree times the identity less the adjacency matrix, so the two spectra are one piece of information and the tree count comes with them; between cospectral regular graphs the critical group is the only exact invariant left. Made by Godsil–McKay switching on four vertices, 705 distinct cospectral pairs of 3- and 4-regular graphs on 12 to 20 vertices give its answer. It separates 181 of 587 quartic pairs and none of 118 cubic ones. And on every one of the 705 the two groups agree everywhere except at the prime two: a switch is a conjugation by a matrix of halves, the group can see it only through its 2-part, and a group whose tree count has fewer than three factors of two cannot see it at all.
No subtraction, and no waiting
The power iteration keeps a positive vector's small entries correct because it never subtracts, and it pays for that in steps. On a Markov chain made of two clusters joined by a coupling ε it needs about 7/ε of them — 6,900 at ε = 10⁻³, 69,000 at 10⁻⁴ — and then stalls at a fixed point of its own rounded arithmetic about u/(4ε) from the answer; at ε = 10⁻⁶ it is still 45 per cent wrong after 200,000. Gaussian elimination needs no waiting and subtracts: on the same chains its error grows like 1/ε, and on a chain whose entries fall to 10⁻¹⁸⁴ it returns the small ones as noise, eight of them negative. The Grassmann–Taksar–Heyman elimination does neither. Every entry of every chain comes back to a relative 10⁻¹⁵, in a third of n³ operations, because the stationary vector is well conditioned under the perturbations it makes.
The stall was in the block
On two dense clusters joined by a coupling ε, the power iteration needs about 7/ε steps and then stalls at about u/(4ε). Aggregation–disaggregation was predicted to need a number of steps independent of ε and to stall at rounding, and with a power step as its smoother it does: fourteen steps at every coupling from a tenth to 10⁻¹², every entry to a relative 10⁻¹⁵. The textbook smoother, block Gauss–Seidel, is faster and stalls at u/ε again — because each block solve forms one minus a self-probability, and a block whose rows sum to 1 − ε is singular to within ε. Solved without that subtraction, the same blocks finish in one to four steps at rounding, and get faster as the coupling weakens.